I will interpret the equivariant cohomology of the cotangent bundle of a flag variety F as the XXX Bethe algebra of a suitable Yangian module. Under this identification the dynamical connection of the XXX model turns into the quantum connection of Braverman-Maulik-Okounkov.
This gives a relation between the quantum integrable chain models and quantum cohomology algebras, the relation discussed in physics literature by Nekrasov and Shatashvili.

Let L be a finite dimensional Lie algebra over a field k of characteristic zero and let S(L) be its symmetric algebra, equipped with its natural Poisson structure. We collect some general facts on the Poisson center of S(L), including some simple criteria regarding its polynomiality, and also on certain Poisson commutative subalgebras of S(L). These facts allow us to give an explicit description for the Poisson center for all complex, nilpotent Lie algebras of dimension at most seven. Among other things, we also provide in each case a polynomial, maximal Poisson commutative subalgebra of S(L), enjoying additional properties. As a by-product we show that a conjecture by Milovanov is valid in this situation. Finally, all these results easily carry over to the enveloping algebra U(L) of L.

Let $X$ be an irreducible smooth projective algebraic curve of genus $g \geq 2$ over the ground field $\mathbb{C}$ and let $G$ be a semisimple simply connected algebraic group. The aim of this talk (joint work with C.S. Seshadri) is to introduce the notion of \textit{semistable and stable parahoric} torsors under a certain Bruhat-Tits group scheme $\mathcal{G}$ and construct the moduli space of semistable parahoric $\mathcal{G}$-torsors; we also identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of $G$. The results give a generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles.

I will discuss the problem of tensor product of semistable and polystable Hitchin pairs in positive characteristics. The problem is closely connected to the work of Serre and others on tensor products of semisimple modules in positive characteristics.

The category of Soergel bimodules is a certain full monoidal subcategory of bimodules over a polynomial ring. Soergel bimodules have many useful and beautiful properties. I will try to explain why Soergel bimodules become even more interesting when one considers them over the real numbers. It turns out that they have all of the properties that one expects from the real cohomology of smooth algebraic varieties! In this way one obtains enough structure to give a proof of Soergel's conjecture (by adapting arguments due to de Cataldo and Migliorini). Corollaries are a proof of the positivity of Kazhdan-Lusztig polynomials and an algebraic proof of the Kazhdan-Lusztig conjecture. (Joint work with Ben Elias.)

Minimal length elements in a conjugacy of a finite Coxeter group were first studied by Geck-Pfeiffer, which have several remarkable properties with respect to conjugation. These properties are very useful in the study of finite Hecke algebras and Delinge-Lusztig varieties. In this lecture, I will talk about the extensions of such properties in the affine Weyl group case and their applications on affine Hecke algebras and affine Delinge-Lusztig varieties. This is based on joint work with X. He.

The goal of this talk is to construct irreducible bounded weight modules for the Lie algebra of vector fields on a torus. These modules have a weight decomposition with finite-dimensional weight spaces and possess the property that the energy operator has spectrum bounded from below. We use vertex operator methods to give explicit realizations for a family of such representations.
The modules in this family are irreducible unless they belong to the chiral de Rham complex, introduced by Malikov, Schechtman and Vaintrob.
This is a joint work with Slava Futorny.

A lot of recent research has sprung out of work towards a conjecture of Guralnick: There is a constant $c$ such that for any finite group $G$ and irreducible, faithful representation $V$ for $G$, the dimension of $H^1(G,V)$ is less than $c$. This conjecture reduces to the case of simple groups. New computer calculations of F. Luebeck and a student of L. Scott show that it is likely that the conjecture is wrong, but if one fixes either the dimension of $V$ or the Lie rank of $G$ then there do exist bounds, due to Parshall--Scott (defining characteristic) and Guralnick--Tiep (cross characteristic). The latter is explicit and the former has now been made explicit by some recent work of A. Parker and myself. There are many more general results, however. I'll give an overview of the area.

We will discuss a few questions of cohomological nature related with current Lie algebras, i.e. algebras formed as the tensor product of a Lie algebra and an associative commutative algebra. We will describe deformations of a certain class of such Lie algebras (generalizing an old result of Lecomte and Roger), give applications to the question of finitness of a number of cohomologically nontrivial modules of a fixed finite dimension, and to some questions about varieties of Lie algebras.

Kostka polynomials are certain family of polynomials indexed by two copies of simple modules of a Weyl group W. They are intimately connected with the (generalized) Springer correspondences, and they admits a characterization by the Lusztig-Shoji algorithm.
In this talk, we reinterpret the Lusztig-Shoji algorithm (of a complex reflection group) in terms of homological algebra. This naturally upgrades Kostka polynomials to a family of indecomposable modules that we call Kostka systems. They give a new characterization and computation method of Kostka polynomials arising from the generalized Springer correspondences in the sense of Lusztig.

Almost every specialist in modern representation theory and related geometry faced the problem of dealing with the affine symmetrizer, the one defined for the affiine Weyl group or its deformation in terms of the corresponding affine Hecke algebra. Its action in DAHA modules appeared a surprisingly deep theory. The classical p-adic spherical functions, the Hall polynomials, the Kac-Moody characters and the affine Demazure characters are among special cases of this new theory. The key results are the proportionality of the DAHA symmetrizer to the Satake map and the clasification of the DAHA coinvariants of higher levels.
The level-one case is directly related to the theory of q-analogs of Mehta-Macdonald integrals with many important aspects in the range from the strong Macdonald conjecture (Fishel, Grojnowski, Teleman) to the theory of global spherical functions, one of the main applications of DAHA so far.

The Baum-Connes conjecture states that a purely topological object of a group coincides with a purely analytical one. The coarse Baum-Connes conjecture is a geometric analogue of the Baum-Connes conjecture. We proved the coarse Baum-Connes conjecture for (the Cayley graph of) relatively hyperbolic groups. In this talk, first I will briefly explain these conjectures and introduce the relatively hyperbolic groups. If time permits, I will also explain the idea of the proof of our result.
This is talk based on the joint work with Shin-ichi Oguni.

I will introduce a construction of a critical cohomological Hall algebra A associated to an arbitrary quiver $Q$, following the work of Kontsevich and Soibelman. This algebra turns out to have several surprising properties: firstly, it is a free commutative algebra. Secondly, the Kac polynomials are the characteristic polynomials of its generators. Finally, despite being defined in terms of vanishing cycles, the underlying vector space of A admits a much simpler description (in fact many such descriptions) in terms of ordinary compactly supported cohomology. I will explain these properties, and why they add up to a new proof of the Kac positivity conjecture.

In the representation theory of reductive groups in positive characteristic, there is a very special class of objects called tilting modules. Two important theorems state that the class of tilting modules is closed under tensor product and restriction to Levi subgroups.
Similarly, on (generalized) flag varieties, there is a special class of geometric objects called parity sheaves. I will explain two newer theorems that are analogues of the tensor product and restriction theorems mentioned above.
The main goal of the talk will be to explain how these two pictures fit together.
This is based on joint work with Daniel Juteau and Geordie Williamson.

(joint work with Francesco Cavazzani) In how many ways can a positive integer be expressed as a repeated sum of elements of a fixed list of positive integers? Generalizing this question, the vector partition function counts in how may ways a vector with integer coordinates can be written as a linear combination with nonnegative integer coefficients of the elements of a list of vectors with integer coordinates. This function is ``piecewise quasi-polynomial'', and its local pieces generate a module over the Laurent polynomials. We describe this and several related modules and algebras. Then we show that these modules and algebras can be``geometrically realized'' as the equivariant K-theory of some manifolds that have a nice combinatorial description.We also propose a more natural and general notion of duality between these modules, which corresponds to a Poincaré duality-type correspondence for equivariant K-theory.

In this talk I will describe geometric realizations of level-one highest weight representations of the Kac-Moody algebra of type $A_n$ on the equivariant cohomology of moduli spaces of rank one framed sheaves on certain root toric stacks. Moreover, I will characterize some (Carlsson-Okounkov type) vertex operators arising from the study of these moduli spaces. These results can be interpreted as AGT type correspondences for abelian (quiver) gauge theories on ALE spaces of type $A_n$. (This talk is based on a joint work with Mattia Pedrini and Richard J. Szabo).

Moment graph techniques have been applied in the study of non-critical blocks of category O for affine Kac-Moody algebras, while in the critical case these methods have not been developed yet. Inspired by the fact that non-critical representations are controlled by the Hecke algebra H, while critical level representations are expected to be governed by the periodic module M, we prove the moment graph analogue of a result by Lusztig which bridges H and M, and believe that it should provide us with new tools to attack the critical level case.

On the affine quantum algebras $U_q$ Drinfeld introduced a coproduct $\Delta$ with values in a completion $V$ of the tensor product $U_q\otimes U(q)$ (in particular it is different from the standard coproduct defined on the generators $E_i, F_i, K_i^{\pm 1}$, which has values in $U_q\otimes U_q$). The Drinfeld coproduct is defined on the Drinfeld generators $X_{i,r}^{\pm}, H^{i,r}, C^{\pm 1}, K_i^{\pm 1}$, but has not yet been proved to be well defined in the general case. In this talk we prove that it is well defined both on the affine quantum algebras and on the quantum affinizations, where it is the only coproduct since the standard one is not defined. The proof depends on the study of some (well defined) derivations of a subalgebra of $V$.

I will review a few general facts about multiplicative actions and outline a proof that algebraic $C^*$-actions on $C^3$ are linearizable. A recent extension to arbitrary fields of characteristic zero is:
Forms of $G_m$-actions on $A^3$ are linearizable. As a corollary one finds that forms of $A^3$ that admit an action of an infinite reductive group are trivial.

Given any affine Lie algebra the associated current algebra is a particular maximal parabolic subalgebra. The category of graded representations of the current algebra have close connections with the category of finite--dimensional representations of the quantum affine algebra.
In this talk, we shall first review these connections. We then discuss a BGG type reciprocity result for current algebras and the relationship with Macdonald polynomials.